3. Equity-Based Guarantees
• Equity-Based Guarantees, namely: -
Fixed Annuities
Variable Annuities
• Variable Annuities
= Unit-linked Investment Product
with given Fund Choices (Unit Trusts / Mutual Funds rather than
Market Indices); and
Investment Allocation / Fund Switching made by Customer;
(may be on the advice of Distributor / Financial Planner);
(can be on an unrestricted manner / frequency);
Plus
Investment Return / Benefit Guarantees offered
at a given fee for the specific guarantee;
which is normally fixed (less dependent on recent investment condition); (and
may not be changeable, regardless of changes in investment conditions in the
future).
3
Risk Management
• Other than Strategic, Operations, Skill-set, Reputation, Litigation,
Counterparty Risks, the Key Risk for managing Variable Annuities is Financial
(Balance Sheet) Risk
• If not Outsourcing the VA management, typical Financial Risk Management
will be: -
Asset Liability
Security Investment ← → Risk Management
Static Hedging ← → Pricing
Trading
Dynamic Hedging ← Desks / Asset ALM Financial → Valuation
Models
Static & Dynamic Hedging ←
Management ↔ → Financial Reporting
Arm
Capital Management (Naked) ← → Capital Planning
4
4. Risk Management
• Financial Modeling to quantify Guarantee Risk: -
Black Box Judgemental
← Historical Data
Quantification Results ← Models ← Assumptions ← Empirical Experiences
← Allowance for Unknown
← Unknown of Unknown
• Nested Stochastic Projections (Number of Scenario Runs)
• Partial Differential Functions to provide Analytical Results
• Heuristic Approach to save Computation Time
• Inter-dependence of Parameters / Path-dependent Scenarios
• Risk on Risk (i.e. Risk Management on Financial Modelling to make sense
for Quantification Results).
5
Risk Management
• Risk Management typically involves Identification, Assessment, Response,
Control, Monitoring
• Practically, Quantification of Risks by Financial Models require
- Understanding of Basics
- Systematic Building of Blocks and Comparing Relativity
- Using Boundary Conditions to help checking reasonableness of Outputs /
Signals
- Identifying Exotic Payoff Movements for small changes in assumptions /
underlier values
- Tradeoff of Risk Alternatives to assess Model Relativity / Consistency
- Reasoned Comparisons of Outputs across different Financial Market
disciplines
- Due Diligence / Disciplined Processes / Actions
6
5. Back to Basics
Assume No Transaction Costs/Lapses/Mortalities
PV (Guarantee Fee Income
≅
Black-Scholes Option Premium (10-
on Single Premium year Vanilla Put with Strike 100)
10-Year GMAB (Point-to-Point))
PV (Guarantee Fee Income
≅ ∑ Discrete Probability Function (K) *
K ≥100
on Single Premium Black-Scholes Option Premium
(10-Year Vanilla Put with Strike K)
10-Year GMAB (Ratchet))
7
Option Pricing
• Under the conditions of Constant Volatility and No Transaction Costs,
Black-Scholes when publishing the option pricing formula, asserted that
the Price of an Option should be the Discounted Value of the cost of
Dynamically Hedging the exposure to Expiration.
• Dynamic Hedging refers to: -
- Delta Hedging of a non-linear position with linear investment(s) of the
underlying
- The deltas of the non-linear position and linear position offset, yielding
a zero delta for the hedged portfolio (π)
- The non-linear position f can generally be expressed in the parabolic
form of the underlying (S): -
f ( S ) = cS + bS + a
2
⎛∂ f ⎞ ⎛ ∂f ⎞
2
where Gamma ⎜ ⎟ = 2 c, Delta ⎜ ⎟ = b, a = f ( S = 0 )
⎝ ∂S ⎠ 2
⎝ ∂S ⎠ 8
6. Dynamic Hedging
δ line
Long Put
f f Short Put f
S S S0 S
π = f −δ * S
Zero Delta
Short δ
f stock S to f
nullify
Delta
S0 S S
9
Dynamic Hedging
• As the underlying (S) changes value, the delta of the non-linear position
changes, but not for the linear instrument. The deltas no longer offset. Thus,
the linear hedge has to be adjusted (increase or decrease exposure) to
restore the delta hedge. The continual adjusting of the linear position to
maintain a zero delta is Dynamic Hedging.
• Writing a VA is equivalent to a Short Put. Short Put creates Negative Gamma
on the non-linear exposure ( curvature opens downward) of the underlying.
• Dynamic Hedging for a Negative Gamma position will lead to Gamma Loss.
10
7. Gamma Loss
At time 0, for the delta hedged portfolio,
∂f
π = f(s ) − δ * S and δ = S = S0
0 0 0 ∂S
At time 1,
a) If S moves up to S + Δ S , π 1 = f ( S + Δ S ) − δ * ( S + Δ S )
0 0 0 0
∂f ∂f
Given Negative Gamma, S =S0 + ΔS < S < s< s0 + Δs < δ
∂S ∂S 0
> π − π = f (S + ΔS ) − δ * (S + Δ S ) − f (S ) − δ * S
[ ( ) ]
1 0 0 0 0 0
= f S + ΔS − f (S ) − δ * ΔS
0 0
⎡ ∂f ⎤
= ⎢ S0 <S <S0 +ΔS − δ ⎥ * Δ S = Negative Value * Δ S = Loss;
⎣ ∂S ⎦
b) If S moves down to S − Δ S , π = f (S − ΔS ) − δ * (S − ΔS )
1 0 1 0 0
∂f ∂f
Given Negative Gamma, S =S0 −ΔS > S >S >S0 −ΔS > δ
∂S ∂S 0
> π − π
1 0
[
= f (S − Δ S ) − f (S ) − δ * ΔS
0 0
]
⎡ ∂f ⎤
= ⎢ S 0 > S > S 0 − Δ S − δ ⎥ * ( − Δ S ) = Positive Value * (- Δ S ) = Loss
⎣ ∂S ⎦
11
Gamma Loss
• For a Negative Gamma Portfolio, it always loses value on delta re-hedge due to
Buy High / Sell Low phenomenon. It never gains it.
• For each re-hedge, the Loss will equal to ½ * Gamma * (Change of Underlying
Value)2.
• The sum of Gamma Loses, till Expiration, is the actual cost of the option (before
Transaction Costs).
• The cash balance on dynamic hedging a Short Put (negative Г) reduces as
follows: -
1 N −1 2 s −s
= ∑ (r − σ 2Δt ) Γ⎛ ⎞ s 2, where r = i + 1 i ,σ = Implied Volatility
2 i =1 i i ⎜ i, s ⎟ i
⎝ i⎠
i s i
i
← Experienced Volatility < Implied Volatility
← Experienced Volatility = Implied Volatility
12
Option Sold Option Expired
8. Pricing Considerations
• As can be seen, at a higher volatility, the underlying will fluctuate more, and the delta
hedge needs to be adjusted more frequently. The cash balance (arisen from the
premium / fee income received) will lose more rapidly when dynamically hedging the
non-linear exposure.
• Thus, the initial key consideration in assessing the sufficiency of VA (i.e. GMB)
charge is: -
- to evaluate the Г of the guarantee with change in value of the underlying till
expiration
-to consider the potential volatility of the underlying (as the ultimate hedging cost
depends on the experienced volatility)
• In addition, for the non-linear exposure f=f(S,σ,r,t)
- By Taylor Expansion
∂f ∂f ∂f ∂f 1 ∂2 f
Δf = Δt + ΔS + Δσ + Δr + (ΔS ) 2 + .....
∂t ∂S ∂σ ∂r 2 ∂S 2
- By Delta Hedging using linear instruments
2
∂f ∂f ∂f ∂f 1 ∂ f 2
Δf − ΔS = 0 = Δt + Δσ + Δr + ( ΔS ) + .....
∂S
2
∂S ∂t ∂σ ∂r 2
Theta Vega Rho └ Gamma Management ┘
Decay 13
Pricing Considerations
• For VA, the non-linear exposure F
~ ~
σ
F = F(f( S i , i , r, t), Si , Ti , Transaction Cost, Profit / Capital Charge)
~
for all Fund Choices Si
where S i is the proxy investment vehicle which can be used to dynamically
~
hedge Si
~ ~
Ti is the period under which S i is selected by the Customer for F to be subject
to.
• The additional pricing considerations or quantifications, therefore, include:
- Basis / Gap Risk ( S i → s i ≅ 10 % extra cost)
~
- Policyholder Behavior (Product Design (MVA) or Fixed Penalty to address?)
-Transaction Costs for Hedging
- Fixed VA Charge vs Variable Theta Decay (Profitability & Profit Variation)
on top of cost considerations for Vega, Rho, Gamma management,
Underlying Volatility implications and Frequency of Hedging.
14
9. Vega
• May estimate the Fair Strike Value of Variance Swap, based on raw option prices.
2
ΔK ΔK
i erT Call ⎡ K ⎤ − 1 ⎜ FT −1⎟
⎛ ⎞
σ =2 ∑
2 i erT put ⎡ K ⎤ + 2 ∑
⎢ ⎥ ⎢ i⎥ ⎜ ⎟
⎣ i⎦ T ⎦ T ⎜K
K ⎣ ⎟
T K ≤F K 2 K >F
K 2 ⎜
⎝ 0
⎟
⎠
i T i i T i
S
where F = e rT , K = i
T i S
0
K is the first strike below the forward F
0 T
• Issue:
- Not sufficient option prices available in the market for longer term T.
- Work best for shorter term T
- Hong Kong: Warrant Transactions (20%-40% daily stock market turnover of US$10-
20bn), but mainly 6-9 months for T. 15
Volatility Surface
• Volatility is not constant (per Black-Scholes) and changes with St, k, T
• When Market falls, price movements are usually sharp. Volatility will shoot
up. Issuer of in-the-money VA (or GMB) will feel a double-hit.
• Implied Volatility
Volabtility Smile
Volatility Skew
Strike Strike
• Volatility skew in equities reflects investors’ fear of market crashes which
would potentially bid up the prices of options at strike below current market
levels.
• Volatility increases with reduction in expiration time. 16
10. Volatility Surface
• The hedge ratio will be off if the non-linear position is dynamically hedged without
incorporating the effect of Volatility Surface into the delta calculation.
• Standard Models to allow these effects include:
- Jump-Diffusion Model adds random / Poisson jumps to the GBM that the
underlying assumes
- Regime-Switching Model probabilistically selects different volatility bases in the
modelling process
- Stochastic Volatility Model models the underlying’s value & its volatility as
stochastic processes
17
Delta-Gamma Hedging
• To subscribe another non-linear instrument (g) in addition to the linear
instrument to hedge against the exposure on non-linear instrument (f).
∂f ∂f 1 ∂2 f 2
Δf = Δt + ΔS + (ΔS )
∂t ∂S 2 ∂S 2
∂g ∂g 1 ∂2g
Δg = Δt + ΔS + (Δ S )2
∂t ∂S 2 ∂S 2
• Hedged Portfolioπ = f + α 1S + α 2 g
⎛ ∂ f ∂ g ⎞
Δ π = ⎜ + α ⎟ Δ t
∂ t ∂ t
2
⎝ ⎠
⎛ ∂ f ∂ g ⎞
+ ⎜ + α + α ⎟ Δ S
∂ S ∂ S
1 2
⎝ ⎠
⎛ ∂ 2 f ∂ 2 g ⎞
⎟ (Δ S )
1 2
+ ⎜
⎜ + α ⎟
∂ S ∂ S 2
2 2
2 ⎝ ⎠
• To Make Terms for Δ S & (Δ S )2 = 0
∂2 f ∂2g ⎛ ∂f ∂g ⎞
+α = 0;⎜ + α1 + α ⎟ = 0
∂S 2 ∂S 2 ⎝ ∂S ∂S ⎠
2 2
∂ g 2
∂ f 2
∂f ∂g ∂2g ∂2 f
α = - ;α 1 = −( − * )
∂S 2 ∂S 2 ∂S ∂S ∂S 2 ∂S 2
2
• f=Negative Gamma → g = Positive Gamma ; with g = Positive Gamma,
direct δ exposure to S reduces 18
11. Delta-Gamma Hedging
• g=Positive Gamma means Long Volatility Option
• Option Illiquid Higher Transaction Costs
• Delta-Gamma Hedging = Dynamic + Static Hedging
• The continuous dynamic hedging (including use of static hedges) will incur
an infinite amount of transaction costs, no matter how small it is.
• In the presence of transaction costs, the absence of arbitrage argument is
invalid, market is incomplete, which leads to many solutions.
• There is no definitive solution on VA management. The success in offering
VA will therefore depend on the availability of the right skill sets, integrated
processes, risk management capabilities by the underwriter to generate
viable and sustained solutions.
19
Hedging Methods
• Finally, the desirable hedging method will, inter alia, depend on Transaction Costs
and the Risk Tolerance / Appetite of the VA underwriter.
• Assume a sale of δ shares of the underlying incurs transaction costs λ/δ/s (λ≥0).
Below are 6 common hedging methods.
a) The Black-Scholes Hedging at Fixed Regular Intervals.
The balance account adjusted on reinstating the target hedge ratio: -
⎡⎛ ∂f ∂f ⎞ ⎛ ∂f ∂f ⎞⎤
⎢⎜ − ⎟ − λ ⎜ − ⎟⎥S
⎜ ⎟ ⎜ ∂S ∂S t ⎟ t+ h
⎢⎝ ∂ S t+
⎣ h ∂S t ⎠ ⎝ t+ h ⎠⎥⎦
b) The Leland Hedging at Fixed Regular Intervals
As per (a) using a modified volatility in the model.
(σ 2
m = σ 2 [1 − λ * Constant * Γ ] )
c) The Delta Tolerance Strategy
∂f
Δ − > h (a given constant) ; Re - hedge to Target Hedge Ratio
∂S
d) The Asset Tolerance Strategy
S (t + Δ t ) − S (t )
> h (a given constant) ; Re - hedge to Target Hedge Ratio
S (t )
20
12. Hedging Methods (con’t)
e) Hedging to a Fixed Bandwidth around Delta
∂f
Δ = ± h per the Delta Tolerance Strategy ;
∂S
Re-hedge to the Hedge Ratio to the nearest boundary of the Hedging Bandwidth
f) The Asymptotic Analysis of Whalley and Wilmott
∂f
1
Δ = ± h ( e − r (T − t ) S Γ 2 ) 3 ;
∂S
Similar to (e), the Size of the Hedging Bandwidth depends on the price of the
underlying and the option gamma
• h depends on the risk aversion of the VA underwriter
• Empirical studies suggest e) & f) are outperforming methods. e) outperforms f) if
the risk tolerance of the VA underwriter is higher.
21
Concluding Remarks
• The above quantification considerations will lead to a relevant Variable Theta Decay
to breakeven the related risks & costs.
• Assuming the risks associated with Policyholder Behavior to be mitigated by
relevant Product Design, it still leaves the issue of Fixed Guarantee Fee for Variable
Theta Decay in the ultimate pricing.
• There are variances in practice within the financial service industry:
Insurer Investment Bank
Use of Capital Capital at Risk(Naked) → Hedging Hedging → Capital for Extremes of Models
r=μ μ −σ
Girsanov’s Theorem r=
θ
Risk Premium Shareholder / Customer to share Customer to bear
• Reasonable Check: How the Customer values the net results across different
financial instruments / products.
22
13. Concluding remarks (con’t)
• The challenge when structuring VA in Hong Kong
US Variable Annuity Hedging via Investment Bank
Long Term Volatility ~ 15% p.a. Implied Volatility ~ 50% p.a.
(Term slope not very deep)
+ Basis / Gap Risk
+ Profit Load
• Implications: -
- Product Design → To find means to stabilize the σ for hedging
→ To reduce anti-selection Policyholder Behavior (Distributor Advice)
- Skill-set → Risk Management and Capital Planning
→ Operations and Integration with Trading Desks
• VA is a very Valued Product for Customer & for Retirement Planning / Use
• Actuary’s Social Responsibility: -
To research, develop, implement and risk manage the Product in a professional manner
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